RESEARCH ARTICLE. Stability of a Ricker-type competition model and the competition exclusion principle

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1 Journal of Biological Dynamics Vol. 00, No. 00, Month 2011, 1 27 RESEARCH ARTICLE Stability of a Ricker-type competition model and the competition exclusion principle Rafael Luís, a Saber Elaydi b and Henrique Oliveira c a Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Tecnico, Technical University of Lisbon, Lisbon, Portugal; b Department of Mathematics, Trinity University, San Antonio, Texas, USA; c Department of Mathematics, Instituto Superior Tecnico, Technical University of Lisbon, Portugal. (January 2010) Our main objective is to study a Ricker-type competition model of two species. We give a complete analysis of stability and bifurcation and determine the center manifolds as well as stable and unstable manifolds. It is shown that the autonomous Ricker competition model exhibits subcritical bifurcation, bubbles, period-doubling bifurcation, but no Neimark-Sacker bifurcations. We exhibit the region in the parameter space where the competition exclusion principle applies Keywords: Ricker competition model; Periodic cycles; Stability; Center manifold; Stable and unstable manifolds; Bifurcation; Exclusion principle. 1. Introduction Discrete models of single species were popularized by the influential paper of Robert May [36] which was published in Nature more than thirty years ago. Twenty years earlier, two significant papers, that marked the beginning of discrete modeling, were published, one by Beverton and Holt [3] and the other by Ricker [39]. Let x t be the population density of species x at time t. Then the ratio f(x t ) = x t+1 /x t is called the fitness function of population x. The intraspecific competition among individuals of species x is classified as either scramble or contest competition [21]. For scramble competition f increases until it reaches a maximum after which f decreases monotonically. This scenario is caused by a phenomenon of overcompensatory in which a large population decreases due to fierce competition on resources and habitat. The Ricker model is an example of a scramble intraspecific competition. For contest competition f is non-decreasing due to increasing utilization of resources [7]. The Beverton-Holt model is an example of contest competition. In the seventies, Hassell [21] and Maynard-Smith and Slatkin [35] integrated both scramble and contest competitions in their models. A readable account on discrete modeling of single species may be found in the paper by Brannstrom and Sumpter [7] where the authors presented the development of several models. Corresponding author. rafael.luis.madeira@gmail.com saber.elaydi@trinity.edu holiv@math.ist.utl.pt ISSN: print/issn online c 2011 Taylor & Francis DOI: / YYxxxxxxxx

2 2 R. Luís et al. The simplest and the most popular competition model was proposed by Lotka [30] and Volterra [44] in the twenties. Though many consider the model of limited practical utility in many complex organisms, it is generally accepted among researchers that the model gave the impetus to theoretical ecology. By now we have a plothera of continuous (differential equations) competition models that have spawned in the twentieth century. However, the discrete counterpart is still lagging behind. Lindstrom [29] has discussed the advantages and disadvantages of utilization discrete models in population dynamics. Though they are simpler than their continuous counterpart to derive, discrete models are much more intractable and less studied. There are two methods of developing discrete models. The first method is based on a discretization of differential equations using either a classical numerical scheme such as Euler and Runga-Kutta methods or a nonstandard method such as Mickens scheme [38]. In the latter method one would derive discrete models from scratch using the underlying biological properties of species. The Leslie-Gower model [28] may be considered as the discrete analogue of the Lotka- Volterra model. A variation of this model was also obtained by Liu and Elaydi [31] via Micken s nonstandard discretization scheme. Numerous papers on discrete competition models have appeared recently and we will mention few of them here, Cushing et al [12 14], Edmunds et al [16], Franke and Yakubu [19], Guzowska, Luís and Elaydi [20], McGehee and Armstrong [37], Clark et al [10], Kulenovic and Nurkanovic [33], Blayneh et al [6], Desharnais et al [15], Jang [25], Roeger [40], Elaydi and Yakubu [18], Blayneh [5], Allen, Kirupaharan, and Wilson [1], Smith [42], Jang and Cushing [26], Yakubu [46], Hirsch [22], Iwata and Takeuchi [24], and AlSharawi and Rhouma [2]. It is noteworthy to mention that in a recent paper Hone et al [23] utilized a Ricker-type model in the study of a predator-prey system. In this paper we adopt the latter approach and focus on an autonomous Rickertype competition model. In section 2 we present the theory associated with the invariant manifolds, namely the center manifolds, the stable manifolds and the unstable manifolds. In section 3 we focus on the study of the stability of the exclusion fixed points and the coexistence fixed point of the autonomous Ricker-type competition model. In section 4 we study the properties of the stable and the unstable manifolds of the fixed points of this model. In section 5 a bifurcation scenario is presented in which we develop a bifurcation diagram in the parameter space. We show the presence of both, period-doubling bifurcation and transcritical bifurcation. Moreover, we found bubbles and subcritical bifurcation. Furthermore, the model under analysis does not have a Neimark-Sacker bifurcation. 2. Invariant Manifolds Let F : R k R k be a map such that F C 2 and F (0) = 0. Then one may write F as a perturbation of a linear map L, F (X) = LX + R(X) (1) where L is a k k matrix defined by L = D(F (0)), R(0) = 0 and DR(0) = 0, where D denotes the Jacobian matrix. Now we will introduce special subspaces of R k, called invariant manifolds [45], that will play a central role in our study of stability and bifurcation.

3 Journal of Biological Dynamics 3 An invariant manifold is a manifold embedded in its phase space with the property that it is invariant under the dynamical system generated by F. A subspace M of R k is an invariant manifold if whenever X M, then F n (X) M, for all n Z +. For the linear map L, one may split its spectrum σ(l) into three sets σ s, σ u, and σ c, for which λ σ s if λ < 1, λ σ u if λ > 1, and λ σ c if λ = 1. Corresponding to these sets, we have three invariant manifolds (linear subspaces) E s, E u, and E c which are the generalized eigenspaces corresponding to σ s, σ u, and σ c, respectively. It should be noted that some of these subspaces may be trivial. The main question here is how to extend this linear theory to nonlinear maps. Corresponding to the linear subspaces E s, E u, and E c, we will have the invariant manifolds the stable manifold W s, the unstable manifold W u, and the center manifolds W c. The center manifold theory [8, 9, 27, 34, 43, 45] is interesting only if W u = {0}. For in this case, the dynamics on the center manifold W c determines the dynamics of the system. The other interesting case is when W c = {0} and we have a saddle. Let E s R s, E u R u, and E c R t, with s + u + t = k. Then one may formally define the above mentioned invariant manifolds as follows: and W s (0) = {x R k F n (x) 0, n } W u (0) = {x R k {q n } n=0, q 0 = x, andf (q k+1 ) = q k, q n 0, n }. It is noteworthy to mention that the center manifold is not unique, while the stable and unstable manifolds are unique. The next result summarizes the basic invariant manifolds theory Theorem 2.1 Invariant manifold theorem: [32, 34] Suppose that F C 2. Then there exist C 2 stable W s and unstable W u manifolds tangent to E s and E u, respectively, at X = 0 and C 1 center manifold W c tangent to E c at X = 0. Moreover, the manifolds W c, W s and W u are all invariant. In appendix A we present the necessary techniques to compute the center manifolds and the stable and unstable manifolds for non-linear maps. We apply these techniques throughout the paper to a Ricker-type competition model. 3. The Ricker competition model The classical Ricker competition model is given by { un+1 = u n exp(k c 11 u n c 12 v n ) v n+1 = v n exp(l c 21 u n c 22 v n ), where the parameters K and L are assumed to be positive real numbers and c ij (0, 1), 1 i, j 2. Letting c 11 u n = x n and c 22 v n = y n yields the system { xn+1 = x n exp(k x n ay n ) y n+1 = y n exp(l y bx n ), (2) where a = c 12 /c 22 and b = c 21 /c 11. Thus a, b > 0. In the language of population dynamics the parameters K and L are known as the carrying capacities of species

4 4 R. Luís et al. y y A B K a L s 1 s 2 L K a s 2 s 1 L b K x K L b x Figure 1. The stability of the exclusion fixed point and the validity of the competition exclusion principle. (i) If 0 < K 2 and L < bk, then (K, 0) is locally asymptotically stable and species y goes extinct. (ii) If 0 < L 2 and L > K/a, then (0, L) is locally asymptotically stable and species x goes extinct. x and y, respectively, while the parameters a and b are the competition parameters. Equation (2) may represented by the map F (x, y) = (xe K x ay, ye L y bx ). Equation (2) has three fixed points, one extinction fixed point (0, 0), and two exclusion fixed points on the axes (K, 0) and (0, L). A possible fourth positive coexistence fixed point (x, y ) may be present. Let us write the map F = (f, g). Then the isoclines are defined as f(x, y) = x and g(x, y) = y. These are the lines ay + x = K denoted by s 1 and y + bx = L denoted by s 2 (see Figure 1 A,B). Moreover, the map F takes a point (x, y) R 2 + lying above (below) s 1 to a point with a smaller (larger) x coordinate. Similarly, the map F takes a point (x, y) R 2 + lying above (below) s 2 to a point with smaller (larger) y coordinate. Note that on the isocline s 1, the population x has no growth, that is x n+1 = x n and on the isocline s 2 the population y has no growth, that is y n+1 = y n. If the two isoclines s 1 and s 2 intersect in the positive quadrant, we will have the positive fixed point (x, y ) = ( K al 1 ab, L bk ). 1 ab There are two cases to consider here: (i) ab < 1 and (ii) ab > 1 (see Figure 2 A,B). The case ab = 1 will be discarded since in this case the two isoclines are parallel and no coexistence fixed point is present. The Jacobian of Equation (2) is given by [ (1 x)e K x ay axe JF (x, y) = K x ay ] bye L y bx (1 y)e L y bx. The Jacobians evaluated at the fixed points are J 0 = JF (0, 0) = [ e K ] 0 0 e L,

5 Journal of Biological Dynamics 5 y y A B K a L s 1 s 2 L K a x,y x,y s 2 s 1 K L b x L b K x Figure 2. Isoclines: (A) The coexistence fixed point of Equation (2) exists if bk < L < K/a and ab < 1. (B) The coexistence fixed point of Equation (2) exists if K < L < bk and ab > 1. In this scenario this a equilibrium is a saddle. J K = JF (K, 0) = [ ] 1 K ak 0 e L bk, J L = JF (0, L) = [ ] e K al 0 bl 1 L and J = JF (x, y ) = [ 1 x ax ] by 1 y. Before present the stability of these fixed points we should mention that H. Smith in [42] have been used monotonicity to prove the global stability of the fixed points of the system { un+1 = u n exp(r(1 u n Bv n )) v n+1 = v n exp(s(1 Cu n v n )), (3) when r, s 1 in which the invariant set is [0, r 1 ] [0, s 1 ]. Notice that by the changes of variables ru = x and sv = y system (3) is equivalent to { xn+1 = x n exp(r x n Br s y n) y n+1 = y n exp(s y n Cs r x n). Consequently, K = r, L = s, a = Br s and b = Cs r. Hence, his global results cover our local analysis when we take the carrying capacities in the unit interval. In the sequel we study the stability of these fixed points Stability of the extinction and exclusion equilibria The eigenvalues of J 0 are e K > 1 and e L > 1 since K, L > 0. Thus (0, 0) is unstable for all K, L > 0.

6 6 R. Luís et al. The eigenvalues of J K are 1 K and e L bk. Thus ρ (J K ) < 1 1 if and only if 0 < K < 2 and L < bk. Thus (K, 0) is locally asymptotically stable if 0 < K < 2 and L < bk. In the parameter space we call this region R 1 (see Figure 3). Below we will prove that when K = 2 this exclusion fixed point is locally asymptotically stable and when L = bk it is unstable. Thus one may define the region R 1 as R 1 = {(K, L) R 2 : 0 < K 2 L < bk} Note that from the inequality K > L/b we obtain K/a > L/ab and consequently L < L/ab < K/a. In Figure 1A we represent the orientation of the isoclines in the phase-space diagram. Similarly, (0, L) is locally asymptotically stable if 0 < L 2 and L > K/a. The region of stability of (0, L) in the parameter space K L is denoted by Q 1 (see Figure 3) and is given by Q 1 = {(K, L) R 2 : 0 < L 2 L > K/a} Note that from the inequality K/a < L it follows that K < K/ab < L/b. In Figure 1B we show the orientation of the isoclines in the phase-space diagram. We now study the stability of the fixed point (K, 0) when ρ (J K ) = 1. This occurs in two cases, the first is when K = 2 and L < bk, in which the eigenvalues are λ 1 = 1 and λ 2 < 1. The second case is when 0 < K < 2 and L = bk, in which λ 1 < 1 and λ 2 = 1. (The case when K = 2 and L = bk at which λ 1 = 1 and λ 2 = 1 will not be investigated in this paper due to the lack of the required techniques). To investigate these cases, we need to use the center manifold theory developed in appendix A. Making the changes of variable u = x K and v = y in Equation (2) we shift the fixed point (K, 0) to (0, 0). Then the new system is given by { un+1 = (u n + K)e u n av n K v n+1 = v n e L v n b(u n +K). (4) Let us now consider the first case, i.e. K = 2 and L < bk. The Jacobian at (0, 0) is now given by J 0 = [ ] 1 2a 0 e L 2b. Consequently, one may write Equation (4) as where and [ ] [ ] [ ] un+1 1 2a un = v n+1 0 e L 2b + v n f(u, v) = (u + 2)e u av 2 + u + 2av g(u, v) = ve L v b(u+2) + e L 2b v. [ ] f(un, v n ), (5) g(u n, v n ) 1 ρ denotes the spectral radius

7 Journal of Biological Dynamics 7 Let v = h(u) with h(u) = αu 2 + βu 3 + O( u 4 ), α, β R. The map h must satisfy the center manifold equation h( u 2ah(u) + f(u, h(u))) + e L 2b h(u) g(u, h(u)) = 0. By Taylor s series this equation is equivalent to ( ) ( ) α e 2b+L α u 2 + be 2b+L α + 4aα 2 β e 2b+L β u 3 + O[u] 4 = 0 Solving the system α e 2b+L α = 0 and be 2b+L α + 4aα 2 β e 2b+L β = 0 yields the unique solution α = 0 and β = 0. Hence h(u) = 0. Consequently, on the center manifold v = 0, the new map f is given by f(u) = u 2ah(u) + f(u, h(u)) = 2 + e u (2 + u). Simple computations show that the Schwarzian derivative of this map at u = 0 is -1. Hence, by [17] the exclusion fixed point (2, 0) is asymptotically stable. We now consider the second case, i.e, 0 < K < 2 and L = bk. After computing the new Jacobian at (0, 0), Equation (4) may be written as where and [ ] un+1 = v n+1 [ ] [ ] 1 K ak un v n f(u, v) = (u + K)e u av (1 K)u + (av 1)K g(u, v) = ve v bu v. [ ] f(un, v n ), (6) g(u n, v n ) In this case computations show that the center manifold is given by h(v) = av (1 ab) av2 K Thus the new map on the center manifold is now ( a( 1 + ab)(4 + a(2 + b( 6 + K)) K) + 2K 2 ) v 3. (7) f(v) = ve v bh(v). (8) ) Computations show that ( f(v) v=0 ) = 1 and ( f(v) v=0 = 2( 1 + ab). Thus by [17] the exclusion fixed point on the center manifold u = h(v) is unstable. More precisely, it is a semi-stable fixed point from the right since 2( 1 + ab) < 0 (see [17, page 31]). We now summarize these remarks in the following result. Theorem 3.1 : hold true: For the autonomous Ricker equation (2), the following statements (1) (0, 0) is unstable. (2) (K, 0) is asymptotically stable if 0 < K 2 and L < bk,

8 8 R. Luís et al. (3) (0, L) is asymptotically stable if 0 < L 2 and L > K/a Stability of the coexistence fixed point: The case ab < 1 ( ) Recall that (x, y ) = K al 1 ab, L bk 1 ab is a coexistence fixed point if bk < L < K/a and ab < 1. (9) In this situation the lines segments s 1 and s 2 intersect as is shown in Figure 2A. In order to find the stability region of (x, y ) we need to find the region where tr(j ) 1 < det(j ) < 1. (Fore more details about this point see [17, page 200]). This is equivalent to det(j ) < 1 det(j ) > tr(j ) 1 det(j ) > tr(j ) 1. If at least one of these inequalities is reversed, then (x, y ) is unstable. Now det(j ) = ab 1 + (1 a)l + (1 b)k (al K)(bK L) ab 1 and tr(j ) = 2(ab 1) + (1 a)l + (1 b)k. ab 1 Consequently, det(j ) < 1 iff det(j ) > tr(j ) 1 iff and finally det(j ) > tr(j ) 1 iff (al K)(bK L) < (1 a)l + (1 b)k, (10) (al K)(bK L) > 0, (11) (al K)(bK L) > 4(ab 1) + 2(1 a)l + 2(1 b)k. (12) Notice that inequality (11) is automatically satisfied by (9). Thus, (x, y ) is locally asymptotically stable if for any fixed a > 0 and b > 0 with ab < 1 the following inequalities hold { (al K)(bK L) < (1 a)l + (1 b)k (al K)(bK L) > 4(ab 1) + 2(1 a)l + 2(1 b)k. (13) The solution of this system leads to the interior of the region identified by the letter S 1 in the (K, L) plane (see Figure 3). The region S 1 is bounded by the lines L = K/a and L = bk and the curve γ 1 (Points on this curve may be include as it is shown bellow). In Appendix B we will show that γ 1 is part of the left branch of the hyperbola defined by the equation (al K)(bK L) = 4(ab 1) + 2(1 a)l + 2(1 b)k.

9 March 30, :25 Journal of Biological Dynamics RL SE HO CRM Journal of Biological Dynamics L 9 L S4 S3 Q4 l3 l2 L=K a S3 Q3 Q2 S2 S4 L=bK S2 l1 Γ3 Γ2 Γ1 L=bK Q1 Q4 L=K a R3 S1 l3 l2 S1 Q3 Q2 l1 R2 R3 R2 Q1 R4 R1 R1 R4 k1 K k2 k3 k1 k2 k3 K Figure 3. The stability regions in the parameter space of the solution of the Ricker competition equation (2) when a > 0 and b > 0 such that ab < 1. In plot on the left the values of the competition parameters are a = b = 0.5 while in plot on the right a = 0.2 and b = 2. In the sequel, we will show that when K and L are on γ1 the coexistence fixed point is asymptotically stable. This happens when ρ(j ) = 1, i.e., λ1 = 1 and λ2 < 1. Note that on the curve γ1 one has L= 2(a 1) + (1 + ab)k ± 2a (2(a 1) + (1 + ab)k)2 + 4a(4(1 ab) + 2(b 1)K bk 2 ). 2a (14) Making the change of variables un = xn x and vn = yn y in Equation (2) we shift the positive fixed point to the origin. Equation (2) is now equivalent to [ ] ][ ] [ ] [ un J11 J12 un+1 f (un, vn ) +. = vn J21 J22 vn+1 g (un, vn ) where f (u, v) = (u + x )ek (u+x ) a(v+y ) x J11 u J12 v, g (u, v) = (v + y )el (v+y ) b(u+x ) y J21 u J22 v, and the Jacobian at (0, 0) is given by [ ] ] [ 1+K a( b+l) a( K+aL) J11 J12 1+ab 1+ab =. 1+ab bk+l J21 J22 b(bk L) 1+ab 1+ab Now we need to diagonalize this matrix. Let us write the diagonal matrix as [ ] ] [ ][ ][ J11 J12 S11 S12 λ1 0 S 11 S 12, = J21 J λ2 S 21 S 22 (15)

10 10 R. Luís et al. where S 11 = (1 + b)k + (1 + a)l +, S 12 = 2b(bK L) (1 + b)k + (1 + a)l, 2b(bK L) λ 1 = 2(ab 1) + (1 b)k + (1 a)l 2(ab 1) + (1 b)k + (1 a)l +, λ 2 = 2(ab 1) 2(ab 1) S 11 = b(bk L), S 12 = (1 + b)k (1 + a)l +, 2 S 21 = b( bk + L), S 22 = (1 + b)k + (1 + a)l +, 2 with = (1 + 2b + (1 4a)b 2 ) K (a(b 1) b 1 + 2a 2 b 2 ) KL + (1 + 2a + a 2 (1 4b)) L 2. Using again a new change of variables u = S 11 z + S 12 w and v = z + w, yields the following system [ ] [ ] [ ] zn+1 λ1 0 zn = + w n+1 0 λ 2 w n [ ] f(zn, w n ), (16) g(z n, w n ) where [ ] [ ] [ ] f(zn, w n ) S11 S12 f(un, v = n ). g(z n, w n ) S 21 S22 g(u n, v n ) Let z = h(w), where h(w) = αw 2 + βw 3 + O(w 4 ). The function h must satisfy the following equation h(λ 2 w + g(h(w), w)) λ 1 h(w) f(h(w), w) = 0. (17) After simplifying this equation, we write the Taylor expansion and then find the values of the constant α and β. Since the computations here are long, we can not find the exact values of α and β. However, we are able to find it numerically. Notice that, the maximum value of the carrying capacity K is given by biggest value on the left branch of hyperbola (B4), namely K max = 2(1 + a a b). 1 ab Notice that depending of the chosen of a and b, K max can be a very large number namely when ab 1. We reduce our analysis when the competition parameters belongs to the interval (0, 2]. Taking randomly the values of a and b in the interval (0, 2] such that ab < 1 and using the value of carrying capacity L given in (14), we vary the carrying

11 11:25 Journal of Biological Dynamics RL SE HO CRM Journal of Biological Dynamics SHH0L March 30, K Figure 4. Part of the values of the Schwarzian derivative of the new map on the center manifold for the coexistence fixed point. In this simulation we use values of the carrying capacity K in the interval (0, 3] and the competition parameters are in the interval (0, 2] such that ab < 1. capacity K in the interval (0, 3] and find numerically the values of α and β. After to do this we compute the value of the Schwarzian derivative of map H(w) = w + g (h(w), w). From our simulations we conclude that the Schwarzian derivative SH(0) < 0, as it is shown in Figure 4. Note that this simulations can be done for a much more large values of a and b. By numerical calculations, on may conclude that on the curve γ1 the coexistence fixed point of Equation (2) must be asymptotically stable. Similar conclusions may be made if we consider the center manifold w = h(z). We now summarize these conclusions in the following theorem. Theorem 3.2 : Suppose that ab < 1 and let S = Int(S1 ) γ1, where Int(S1 ) denotes the interior of S1. Then the coexistence fixed point (x, y ) = ( al K bk L, ab 1 ab 1 ) of the Ricker equation (2) is asymptotically stable if 4(ab 1) + 2(1 a)l + 2(1 b)k (al K)(bK L) < (1 a)l + (1 b)k. Equivalently, the coexistence fixed point is asymptotically stable if (K, L) S. 4. The stable and unstable manifolds In this section we study (via numerical computations) a celebrated scenario in classic competition theory, the saddle exclusion case (or equivalently, a stable and unstable manifolds). A general two-dimensional map has a stable and an unstable manifolds when the

12 12 R. Luís et al. following conditions, in the Trace-Determinant plane, are satisfied { det(j ) < tr(j { ) 1 det(j det(j ) > tr(j ) 1 ) > tr(j ) 1 det(j ) < tr(j ) 1 det(j ) < (tr(j )) 2 4 det(j ) < (tr(j )) 2 det(j 4 ) > 1 det(j det(j ) > tr(j ) > tr(j ) 1. (18) ) 1 det(j ) > 1 For more details about these conditions see [17, page 205]. Hence we have two scenarios to consider: (i) ab > 1 in which the winner depends on initial conditions and (ii) ab < 1 where we have the presence of both locally asymptotically stable cycles and unstable fixed points Case (i): ab > 1 In model (2) the saddle scenario occurs when one has a coexistence equilibrium such that which implies that al > K and bk > L, (19) ab > 1. (20) We now determine, in the parameter space, the region where relation (18) is satisfied. Direct computations show that det(j ) > tr(j ) 1 is equivalent to (al K)(bK L) < 0, which is impossible by (19). Hence there are two systems in (18) that leads to an empty region. Analogously, det(j ) < tr(j ) 1 is the region in the (K, L) plane between the two lines L = K/a and L = bk, i.e., assumption (19). Notice that by (20) one has b > 1/a, and consequently bk > K/a. The inequality det(j ) > tr(j ) 1 leads to 2(1 a)l + 2(1 b)k (al K)(bK L) + 4(ab 1) > 0 Notice that by (B4) the second degree equation 2(1 a)l + 2(1 b)k (al K)(bK L) + 4(ab 1) = 0 (21) represents an hyperbola in (K, L) plane (for more details about this hyperbola see appendix B). Hence the system { det(j ) < tr(j ) 1 det(j ) > tr(j ) 1 (22) is satisfied whenever K and L are between the lines L = K/a and L = bk and the right branch of hyperbola (21). Relation det(j ) > 1 is equivalent to (1 a)l + (1 b)k (al K)(bK L) > 0.

13 Journal of Biological Dynamics 13 This is the region in the first quadrant outside the right branch of hyperbola (1 a)l + (1 b)k (al K)(bK L) = 0 (23) which passes in the origin. The inequality det(j ) < (tr(j )) 2 4 leads to the following relation (4a 2 b (a + 1) 2 )L 2 + 2(1 + a + b ab 2(ab) 2 )KL + (4ab 2 (b + 1) 2 )K 2 < 0 (24) Notice that the second degree equation (4a 2 b (a + 1) 2 )L 2 + 2(1 + a + b ab 2(ab) 2 )KL + (4ab 2 (b + 1) 2 )K 2 = 0 represents a conic, in the (K, L) plane, known as a pair of imaginary lines intersecting in a real point (see for instance [4]), provided that the test condition are 4ab 2 (b + 1) a + b ab 2(ab) a + b ab 2(ab) 2 4a 2 b (a + 1) = 0 and ( ) 4ab 2 (b + 1) a + b ab 2(ab) a + b ab 2(ab) 2 4a 2 b (a + 1) 2 = 4ab( 1 + ab) 3 < 0 This point is precisely the origin because the equations of these two lines are L = m ± K where m ± = 1 + a + b ab 2a2 b 2 ± 2 ab( 1 + ab) 3 ((1 + a) 2 4a 2. b) Hence, the system det(j ) < (tr(j )) 2 4 det(j ) > 1 det(j ) > tr(j ) 1 (25) represents the region in the (K, L) plane outside both hyperbolas (21) and (23) and between the two lines L = m ± K. Numerical computations show that when a > 1 and b > 1 (which implies ab > 1) one has m + > b > 1/a > m. So assuming this restriction on the competition parameters and under hypothesis (19) system (25) has an empty solution. Consequently, if a > 1 and b > 1 relation (18) is equivalent to system (22). Hence, the saddle region is enclosed by the two lines L = K/a and L = bk and the right branch of hyperbola (21). In Figure 5 is depicted in the parameter space (K, L) this region when a = 2 and b = 1.5. Notice that if we assume that either a < 1 or b < 1 such that ab > 1, then the saddle region is more involved, namely it contains the solution of both systems (22) and (25).

14 14 R. Luís et al. L L bk det J tr J 1 Z L K a Figure 5. The saddle region Z, in the parameter space K and L, when a = 2 and b = 1.5. K Now let us take a > 1 and b > 1 such that (K, L) S. Following the same techniques as in section (3.2) we find that, locally, the stable manifold of the coexistence fixed point is given by and the unstable manifold is W s = {(z, w) R 2 : w = α 1 z 2 + β 1 z 3, α 1, β 1 R}, W u = {(z, w) R 2 : z = β 2 w 2, β 2 R}.p Due the big size of the formulas for the constants α 1, β 1 and β 2 we omit them here. In the original coordinates the values of z and w are given by z = S 22(x x ) S 12 (y y ) S 11S 22 S 21S 12 and w = S 11(y y ) S 21 (x x ) S 11S 22 S 21S 12, where S ij are the entries of the matrix S determined in the previous section Case (ii): ab < 1 By (9) it follows that det(j ) < tr(j ) 1 is impossible. Hence the first system in (18) leads to an empty region. In previous subsection we determine the inequality det(j ) < (tr(j )) 2, 4 which leads to relation (24). We claim that when ab < 1 the first member of (24) is negative. In order to show that, let us assume temporally that L = mk for some m > 0. Hence, relation (24) is equivalent to K 2 u(m) < 0, where u(m) = (4a 2 b (a + 1) 2 )m 2 + 2(1 + a + b ab 2(ab) 2 )m + 4ab 2 (b + 1) 2.

15 Journal of Biological Dynamics 15 L L K a a 1 a ba 2 a Z L bk S 1 det J tr J 1 b 1 b ab 2 b K Figure 6. The saddle region Z, in the parameter space K and L, when a = b = 0.5. Solving the equation u(m) = 0 one has the following two values m = (1 + a + b ab 2(ab)2 ) ± 2 ab(1 ab) 3 i (1 + a) 2 + 4a 2, i = 1. b Now we show that the coefficient of m 2 is a negative number. So 4a 2 b (a + 1) 2 = 4aab (a + 1) 2 < 4a (a + 1) 1 = (a 1) 2 < 0 (since ab < 1). Hence, the function u is a parabola, concave down, with no real zeros. Consequently, relation (24) is satisfied. Because det(j ) > tr(j ) 1 is automatically verified and by the fact that det(j ) < (tr(j )) 2 4 is always true, it follows that the system det(j ) < (tr(j )) 2 4 det(j ) > tr(j ) 1 det(j ) > 1 leads to the same region in the parameter space than det(j ) < (tr(j )) 2 4 det(j ) > tr(j ) 1. det(j ) > 1 Therefore, relation (18) leads to { det(j ) > tr(j { ) 1 det(j det(j ) < tr(j ) 1 ) > tr(j ) 1 det(j. ) > 1 This leads to the region Z identified in Figure 6. Note that Z = i 2 S i (in section

16 16 R. Luís et al. 5 we will give more details about the regions S i, i 2). Before end this subsection we remark that the determination of the saddle and unstable manifolds follows the guidelines as above The exclusion fixed point We now determine the region, in the parameter space, where the exclusion fixed point (K, 0) of Equation (2) has stable and unstable manifolds. This set is given by Z K = {(K, L) R 2 : K > 2 L < bk}. Let (K, L) Z K. Then similar techniques as before lead us to find locally the stable manifold and the unstable manifold of the exclusion fixed point (K, 0). Similarly, in the set W s K = {(x, y) R 2 : y = 0} W u K = {(x, y) R 2 : x = K} Z L = {(K, L) R 2 : K > 0 L > 2 L > K/a} the exclusion fixed point (0, L) has the stable manifold (locally) and the unstable manifold (locally) W s L = {(x, y) R 2 : x = 0} W u L = {(x, y) R 2 : y = L}. 5. Bifurcation scenarios In the absence of species y the dynamics of species x is governed by the onedimensional Ricker equation x n+1 = x n e K x n, n Z +. (26) Equation (26) has a globally asymptotically stable fixed point when 0 < K < 2. At K = k 1 = 2, a period-doubling bifurcations occurs. At the bifurcation point k 1 = 2, an asymptotically stable 2 periodic cycle {x 0, x 1 } is born. The two points x 0, x 1 satisfy the equations x 1 = x 0 e K x0 and x 0 = x 1 e K x1. By the linearization principle the stability of this 2 periodic cycle can be seen from the product of the derivatives of the map (26) evaluated at x 0 and x 1. This product is less than one in absolute value, i.e., 1 i=0 1 x i < 1 if k 1 < K < k 2, where k At K = k 2, a new period-doubling bifurcation occurs. Then there exists k 3 greater than but near k 2 such that a new 4 periodic cycle is asymptotically stable if k 2 < K < k 3. This period-doubling scenario continues. So there exist two bifurcations points k j

17 Journal of Biological Dynamics 17 and k j+1 for a specific integer j such that the r periodic cycle {x 0,..., x r 1 }, where r = 2 j, satisfy the relation r 1 1 x i < 1. (27) i=0 The r periodic cycle {x 0,..., x r 1 } yields an exclusion r periodic cycle C x r = {(x 0, 0), (x 1, 0),..., (x r 1, 0)} (28) of the competition Ricker model (2). The Jacobian of C x r evaluated along the periodic orbit is given by the following 2 2 matrix 0 JF (x i, 0) = r 1 ( r 1 i=0 (1 x ) i) J 12 0 e rl b r 1. i=0 xi Its eigenvalues are λ 1 = r 1 i=0 (1 x i) and λ 2 = e rl b r 1 x i=0 i = er(l bk). Using the hypothesis L < bk yields λ 2 = λ 2 < 1 and from (27) it follows λ 1 < 1. Thus Cr x is asymptotically stable. Note that if L = 0 one has λ 2 < 1. This implies that the sequence of parameters {k j } on the K axis follows the one-dimensional case. That is k 1 = 2, k , k , etc. We now summarize the above discussion Theorem 5.1 : Let 0 < L < bk. Then the cycle Cr x, defined in (28), of Equation (2), is asymptotically stable. In [17] the author presents a complete study of the main types of bifurcations, for two-dimensional systems. When the Jacobian has an eigenvalue equal to one, either the saddle-node bifurcation, the pitchfork bifurcation or the transcritical bifurcation can occur. In the Trace-Determinant plane (T-D), this is equivalent to saying that we cross the line det(j ) = tr(j ) 1 from the stability region. The period-doubling bifurcation occurs when the Jacobian has an eigenvalue equal to -1. In the T-D plane this occurs as we cross the line det(j ) = tr(j ) 1 from the stability region. When the Jacobian has a pair of complex conjugate eigenvalues of modulus 1, we have the Neimark-Sacker bifurcation. This happens in the T-D plane when det(j ) = 1 and 2 < tr(j ) < 2. For details on bifurcation in higher dimension see for example [45]. Now we are in a position to provide a deeper explanation of Figure 3. Note that the coexistence fixed point (x, y ) is asymptotically stable if (K, L) Ŝ. When L = bk, the Jacobian of Equation (2) has an eigenvalue equal to one. For the map f defined in (8) one has f f 2 f (0) = 1, (0) = 0 and (0) 0. x K x2 Hence a transcritical bifurcation occurs when L = bk, where the coexistence fixed point (x, y ) = (K, 0), the exclusion fixed point on the x axis. When (K, L)

18 18 R. Luís et al. A B x n k C x n y n y n k D k k Figure 7. The presence of subcritical bifurcation in the autonomous Ricker type competition model (2) crosses the line L = bk to region R 1, the branch of equilibria (x, y ) transcritically bifurcates with the branch of exclusion equilibria (K, 0), while (x, y ) moves from the first quadrant into the fourth (or second) quadrant, where it becomes ecologically irrelevant. Moreover, stability exchanges from one branch to the other. Similarly, if L = K/a, the coexistence fixed point undergoes a transcritical bifurcation. Equation (2) has a period-doubling bifurcation when we have equality in relation (12). This is represented by the curve γ 1 in Figure 3. Consequently, as K and L passes the curve γ 1 the coexistence fixed point undergoes a period-doubling bifurcation into a coexistence 2 periodic cycle. Thus in region S 2 Equation (2) has one unstable fixed point and one asymptotically stable coexistence 2 periodic cycle. When K and L passes the line L = bk from region S 2 to region R 2, the coexistence 2 periodic cycle bifurcates (transcritical). Computations shows that this new 2 periodic cycle is an exclusion cycle on the x axis. If, however, we move K and L from R 2 to S 2, then the exclusion 2 periodic cycle undergoes a transcritical bifurcation into a coexistence 2 periodic cycle. Another period-doubling bifurcation appears in the exclusion fixed point if the parameters K and L move from region R 1 to region R 2. Thus if the parameters K and L are in region R 2, Equation (2) possesses an asymptotically stable exclusion 2 periodic cycle on the x axis. Similar analysis can be taken if the parameters are in region Q 2. The coexistence 2 periodic cycle undergoes a period-doubling bifurcation when the parameters pass the curve γ 2. Thus in region S 3, this coexistence 2 periodic cycle becomes unstable and a new asymptotically stable 4 periodic cycle is born. This new cycle undergoes a transcritical bifurcation to an asymptotically stable exclusion 4 periodic cycle on the x axis whenever the parameters moves from region S 3 to region R 3. We also have a period-doubling bifurcation in the exclusion 2 periodic cycle if we change the parameters from region R 2 to region R 3. Thus in region R 3, Equation (2) has an asymptotically stable exclusion 4 periodic cycle. The same happens in the y axis if the parameters change from region S 3 to region Q 3. This period-doubling bifurcation route to chaos is reminiscent of the dynamics exhibited by the one-dimensional Ricker-map. A different scenario appears if the relation between L and K obey the rule

19 Journal of Biological Dynamics 19 L = α 1 K + α 2, α 2 > 0 and ε < α 1 < ε, for a small ε > 0. We call this scenario the bubble scenario. This occurs if one passes from zone S i+1 to the stability region S i and enter again in the stability region S i+1. In this scenario, if we draw the bifurcation diagram in the (K, x) plane we find bubbles. In Figure 7 we present two scenarios. In cases A and B we vary K and fix L = 2.1, and let a = b = 0.5. This results in the presence of one bubble in plot A. This phenomenon happens because for values of K 1.05, Equation (2) has one attracting 2 periodic cycle on the y axis (see plot B). Thus x n = 0 and y n oscillates between and as n goes to infinity. At K 1.05, the exclusion cycle on the y axis bifurcates (transcritical) and the fixed point 0 bifurcates (period-doubling). This implies that the coexistence 2 periodic cycle in the x axis is born. Here we see the bubble in plot A and a 2 periodic cycle in plot B. For values 1.45 K 1.78, Equation (2) has a coexistence fixed point. This observation implies that at K 1.45 the 2 periodic cycle in turn will undergo a bifurcation and return to a stable equilibrium 1. At K 1.78 a new period-doubling bifurcation occurs, and then the coexistence fixed point bifurcates into a coexistence 2 periodic cycle. This is clearly shown in plot A and plot B. In cases C and D we fix L = 2.6. For values of K 1.3 the equation has an exclusion 4 periodic cycle on the y axis. As K increases we enter in the zone where we have a coexistence 4 periodic cycle. Here we see two bubbles in plot C and a coexistence 4 periodic cycle in plot D. Both cases lead to a 2 periodic cycle. The Neimark-Sacker bifurcation starts when det(jf (x, y )) = 1 and 2 < tr(jf (x, y )) < 2, i.e, when and (1 a)l + (1 b)k = (al K)(bK L) (29) 0 < (1 a)l + (1 b)k < 4(1 ab). (30) Inequalities (30) are satisfied whenever K and L belongs to the region enclosed by the positive axes and the line (1 a)l = (1 b)k +4(1 ab). Direct computations show that this line( does not intersect ) the hyperbola (29). On the other hand the vertices (0, 0) and 2 1+a 1 ab, 2 1+b 1 ab of hyperbola (29) are outside this triangle. Hence Equation (2) has no Neimark-Sacker bifurcation. 6. Discussion Competition models have been investigated by many authors and we listed few in the references. Here we consider a mathematical model (2) in which the intraspecific competition parameters are scaled to 1. This in effect reduces the number of parameters by two and we only have four parameters. The main contribution in this paper is to analyze in depth the bifurcation scenario of the main parameters K and L, the carrying capacities of species x and y, respectively. So Figure 3 shows that our model exhibits period-doubling bifurcation of the coexistence equilibrium point in regions S i, i = 1, 2, 3,.... The regions S i, i = 1, 2, 3,... are bounded by the lines L = bk, and L = K/a and the curve γ i. Each curve γ i is a segment of a hyperbola, which we determine 1 Actually this phenomenon is not a reverse period doubling bifurcation. It is a subcritical bifurcation. For more details about this phenomenon in population models see [11]

20 20 REFERENCES explicitly for i = 1. Hence we conclude that the coexistence of two species is possible only iff (K, L) S i, i = 1, 2, 3,.... The two species may coexist in a variety of forms: an equilibrium state, oscillatory periodic cycles of period 2 n, and eventually chaotic. The regions R i and Q i, i = 1, 2, 3,... in the parameter space K L are regions where the competition exclusion principle is valid. In regions R i, species y goes to extinction while species x tends to its carrying capacity K or oscillate periodically with periods 2 n and eventually enters a chaotic region. Similar conclusions may be stated for regions Q i, in which species x goes to extinction while species y survives. Finally, our analysis here of the period-doubling bifurcation scenario and identification of the regions in which the competition exclusion principle is valid may be extended to other competition models. In particular, we believe that our analysis here may be applied to many population models with rich dynamics such as the logistic competition model in [20]. References [1] L. Allen, N. Kirupaharan, and S. Wilson, SIS Epidemic Models with Multiple Pathogen Strains, Journal of Difference Equations and Applications, Vol 10, No 1 (2004), pp [2] Z. AlSharawi and M. Rhouma, Coexistence and extinction in a competitive exclusion Leslie/Gower model with harvesting and stocking, Journal of Difference Equations and Applications, Vol 15, No (2009), pp [3] R. Beverton, and S. Holt, On the Dynamics of Exploited Fish Populations, Fishery Investigations Series II, Volume XIX (1957), Ministry of Agriculture, Fisheries and Food [4] Beyer, W. H., CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp , [5] K. Blayneh, Hierarchical Discrete Models of Competition for a Resource, Journal of Difference Equations and Applications, Vol 10, No 3 (2004), pp [6] K. Blayneh, A. Yakubu, and J. Mohammed-Awel, Discrete hierarchical competition with reward and cost of dispersion, Journal of Difference Equations and Applications, Vol 15, No 4 (2009), pp [7] A. Brannstrom and D. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B (2005), p.p [8] X. Cabre, E. Fontich and R. de la LLave, The parameterization method for invariant manifolds: Manifolds associated to non-resonant subspaces,software for Railways Control and Protection Systems. EN 50128, 1995 [9] J. Carr, Application of Center Manifolds Theory, Applied Mathematical Sciences, 35, Springer-Verlag, New York, [10] D. Clark, M. Kulenovic and J. Selgrade, Global asymptotic behavior of two-dimensional difference equation modelling competition, Nonlinear Anal., Vol. 52 (2002), pp [11] J. Cushing, et al, Chaos in Ecology: Experimental Nonlinear Dynamics, Academic Press, 2003 [12] J. Cushing, S. Henson, and C. Blackburn, Multiple mixed-type attractors in a competition model, Journal of Biological Dynamics, Vol 1, No 4 (2007), pp [13] J. Cushing, S. Henson, and L. Roeger, Coexistence of competing juvenile-adult structured populations, Journal of Biological Dynamics, Vol 1, No 2 (2007), pp [14] J. Cushing, S. Levarge, N. Chitnis, and S. Henson, Some discrete competition models and the competitive exclusion principle, Journal of Difference Equations and Applications, Vol 10, No (2004), pp [15] R. Desharnais, J. Edmunds, R. Costantino, and S. Henson, Species competition: uncertainty on a double invariant loop, Journal of Difference Equations and Applications, Vol 11, No 4-5 (2005), pp [16] J. Edmunds, J. Cushing, R. Constantino, S. Henson, B. Dennis and R. Desharnais, Park s Tribolium competition experiments: a non-equilibrium species coexistence hypothesis, J. Anim. Ecol., Vol. 72 (2003), [17] S. Elaydi, Discrete Chaos: With Applications in Science and Engineering, Chapman and Hall/CRC, Second Edition, [18] S. Elaydi and A. Yakubu, Global Stability of Cycles: Lotka-Volterra Competition Model With Stocking, Journal of Difference Equations and Applications, Vol 8, No 6 (2002), pp [19] J. Franke and A. Yakubu, Exclusionary population dynamics in size-structured discrete competitive systems, Journal of Difference Equations and Applications, Vol 5(3)(1999), pp [20] M. Guzowska, R. Luís and S. Elaydi, Bifurcation and invariant manifolds of the logistic competition model, Journal of Difference Equations and Applications, to appear. [21] M. Hassell, density dependence in single species populations, J. Anim. Ecology, Vol. 45 (1975), p.p [22] M. Hirsch, On existence and uniqueness of the carrying simplex for competitive dynamical systems, Journal of Biological Dynamics, Vol 2, No 2 (2008), pp [23] A. Hone, M. Irle and G. Thurura, (2010) On the Neimark-Sacker bifurcation in a discrete predatorprey system, Journal of Biological Dynamics, First published on: 17 February 2010 (ifirst)

21 Journal of Biological Dynamics 21 [24] S. Iwata and Y. Takeuchi, The relationship between endophyte transition and plant species coexistence, Journal of Biological Dynamics, Vol 3, No 4 (2009), pp [25] S. Jang, A discrete, size-structured chemostat model of plasmid-bearing and plasmid-free competition, Journal of Difference Equations and Applications, Vol 11, No 7 (2005), pp [26] S. Jang and J. Cushing, Dynamics of hierarchical models in discrete-time, Journal of Difference Equations and Applications, Vol 11, No 2 (2005), pp [27] A. Kelly, The stable, center-stable, center-unstable and unstable manifolds, J. Differential Equations, 3 (1967), pp [28] P. Leslie and J. Gower, The properties of a stochastic model for two competing species, Biometrica, Vol 45(1958), pp [29] T. Lindstrom, Discrete models and Fisher s maximum principle in ecology, Proceedings of the fourth international conference on dynamical systems and difference equations (2002), pp [30] A. Lotka, Elements of physical biology, Williams & wiliams, 1925 [31] P. Liu and S. Elaydi, Discrete competitive and cooperative models of Lotka-Volterra type, Journal of Computational Analysis and Appllications, Vol 3, No. 1 (2001), pp [32] N. Karydas and J. Schinas, The center manifold theorem for a discrete system, Applicable Analysis, 44 (1992), pp [33] M. Kulenovic and M. Nurkanovic, Asymptotic behavior of a two-dimensional linear fractional system of a difference equation, Radovi Mathematicki, Vol. 11 (2002), pp [34] J. Marsden and M. McCracken, The hopf bifurcation and its application, Springer-Verlag, New York, [35] J. Maynard-Smith and M. Slatkin, The stability of predator-prey systems, Ecology, Vol 54 (1973), pp [36] R. May, Simple mathematical models with very complicated dynamics, Nature, Vol. 261 (1976), pp [37] R. McGehee and R. Armstrong,Some mathematical problems concerning the ecological principle of competitive exclusion, J. Diff. Eqns., Vol 23 (1977), pp [38] R. Mickens, Discretizations of nonlinear differential equations using explicit nonstandard methods, J. Comp. Appl Math., 110 (1999), [39] E. Ricker, Handbook of computation for biological statistics of fish populations, Bulletin 119 of the Fisheries Resource Board, Canada, Ottawa, 1958 [40] L. Roeger, A nonstandard discretization method for Lotka-Volterra models that preserves periodic solutions, Journal of Difference Equations and Applications, Vol 11, No 8 (2005), pp [41] W.Rudin, Principles of Mathematical Analysis, McGraw - Hill, Third Edition, [42] H. Smith, Planar competitive and cooperative difference equations, Journal of Difference Equations and Applications, Vol 3, No 5-6 (1998), pp [43] A. Vanderbauwhede, Center Manifolds, Normal Forms and Elementary Bifurcations, In Dynamics Reported, Vol. 2 (1989) [44] V. Volterra, Variations and fluctuations of the number of individuals in animal species living together in animal ecology, Editor: R. N. Chapman, McGraw-Hill, [45] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, 2003 [46] A. Yakubu, Allee effects in a discrete-time SIS epidemic model with infected newborns, Journal of Difference Equations and Applications, Vol 13, No 4 (2007), pp Appendix A. Center manifolds and the stable and unstable manifolds In this section we give the appropriate tools that allow us, to compute analytically, the center manifolds and the stable and unstable manifolds for any non linear map. We use the terminology and the notation present in section 2. A.1. Center manifolds Let us first focus on the case when σ u =. Hence the eigenvalues of L are either inside the unit disk or on the unit disk. By suitable change of variables, one may represent the map F as the following system of difference equations { xn+1 = Ax n + f(x n, y n ) y n+1 = By n + g(x n, y n ). (A1) First we assume that all eigenvalues of A t t are on the unit circle and all the eigenvalues of B s s are inside the unit circle, with t + s = k. Moreover, f(0, 0) = 0, g(0, 0) = 0, Df(0, 0) = 0 and Dg(0, 0) = 0.

22 22 R. Luís et al. E s i W s W c E c ii W c W s E c E s Figure A1. Stable and center manifolds. In Figure (i) one has σ(a) = σ c and σ(b) = σ s while in Figure (ii) one has σ(a) = σ s and σ(b) = σ c. Since W c is tangent to E c = {(x, y) R t R s y = 0}, it may be represented locally as the graph of a function h : R t R t such that W c = {(x, y) R t R s y = h(x), h(0) = 0, Dh(0) = 0, x < δ for a sufficiently small δ}. Furthermore, the dynamics restricted to W c is given locally by the equation x n+1 = Ax n + f(x n, h(x n )), x R t (A2) The main feature of Equation (A2) is that its dynamics determine the dynamics of Equation (A1). So if x = 0 is a stable, asymptotically stable, or unstable fixed point of Equation (A2), then the fixed point (x, y ) = (0, 0) of Equation (A1) possesses the corresponding property. To find the map y = h(x), we substitute for y in Equation (A1) and obtain { xn+1 = Ax n + f(x n, h(x n )) y n+1 = h(x n+1 ) = h(ax n + f(x n, h(x n ))). (A3) But y n+1 = By n + g(x n, y n ) = Bh(x n ) + g(x n, h(x n )). (A4) Equating (A3) and (A4) yields the center manifold equation h[ax n + f(x n, h(x n ))] = Bh(x n ) + g(x n, h(x n )) (A5) Analogously if σ(a) = σ s and σ(b) = σ c, one may define the center manifold W c, and obtain the equation where x = h(y). y n+1 = By n + g(h(y n ), y n ),

23 A.2. An upper (lower) triangular System Journal of Biological Dynamics 23 In working with concrete maps, it is beneficial in certain cases to deal with the system without diagonalization. Let us now consider the case when L is a block upper triangular matrix ( ) xn+1 = y n+1 There are two cases to consider: ( ) ( ) A C xn + 0 B y n ( ) f(xn, y n ), (A6) g(x n, y n ) (1) Assume that σ(a) = σ s, σ(b) = σ c, and σ u =. The matrix L can be block diagonalizable. Hence there exists, a nonsingular matrix P of the form P = [ ] P1 P 3 0 P 2 such that Let [ ] [ ] A B A 0 = P P 0 C 0 B 1. ( ) x = P y ( u v ). (A7) Then x = P 1 u + P 3 v, and y = P 2 v. Thus one has ( ) ( ) ( ) ( ) un+1 A 0 un = + P v n+1 0 B v 1 f(p1 u + P 3 v, P 2 v). (A8) n g(p 1 u + P 3 v, P 2 v) Applying the center manifold theorem to Equation (A8) yields a map u = h(v) with h(0) = 0 = h (0). Moreover, the dynamics of Equations (A8) is completely determined by the dynamics of the equation v n+1 = Bv n + P 2 g(p 1 h(vn ) + P 3 v n, P 2 v n ), where P 1 and P 3 are elements of the matrix We now have the relation Hence x = h(y), where h is given by [ ] P1 P 1 P3 =. 0 P2 u = P 1 x P 2 P 3 P2 y = h( P 2 y). h(y) = P 3 P2 y + 1 P 1 h 2 ( P 2 y). Notice that Dh(0) = P 3 P2 I, where I is the identity matrix. (2) Assume that σ(a) = σ c, σ(b) = σ s, and σ u =. We start from Equation (A8) and apply the center manifold theorem to obtain a map v = h(u) with